Quantum Tunneling and Back Reaction

We give a correction to the tunneling probability by taking into account the back reaction effect to the metric of the black hole spacetime. We then show how this gives rise to the modifications in the semiclassical black hole entropy and Hawking temperature. Finally, we reproduce the familiar logarithmic correction to the Bekenstein-Hawking area law. In 1975, Hawking discovered [1] the remarkable fact that black holes, previously thought to be completely black regions of spacetime from which nothing can escape, actually radiate a thermal spectrum of paticles and that the temperature of this radiation depends on the surface gravity of the black hole by the relation TH = K0 2π . This discovery was consistent with an earlier discovery [2, 3, 4, 5] of a connection between black holes and thermodynamics which revealed that the entropy of a black hole is proportional to the surface area of its horizon, SBH = A 4 . From this, using the second law of thermodynamics dM = THdSBH , the temperature of the black hole can be calculated. This is actually due to an analogy between the second law of thermodynamics and the black hole equation dM = K0 8π dA. These relations are all based on classical or semiclassical considerations. It is possible to include quantum effects in this discussion of Hawking radiation. Using the conformal anomaly method the modifications to the spacetime metric by the one loop back reaction was computed [6, 7]. Later it was shown [8, 9] that the Bekenstein-Hawking area law was modified, in the leading order, by logarithmic corrections. Similar conclusions were also obtained by using quantum gravity techniques [10, 11, 12]. Likewise, corrections to the semiclassical Hawking temperature were derived [7, 8]. A particularly useful and intuitive way to understand the Hawking effect is through the tunneling formalism as developed in [13]. The semiclassical Hawking temperature is very simply and quickly obtained [13, 14] in this scheme by exploiting the form of the semiclassical tunneling rate. A natural question that arises in this context is the feasibility of this approach to include quantum corrections. Although there have been sporadic attempts [15, 16] a systematic, thorough and complete analysis is still lacking. In this paper we compute the corrections to the semiclassical tunneling rate by including the effects of self gravitation and back reaction. The usual expression found in [13], given in the Maxwell-Boltzmann form e − E TBH , is modified by a prefactor. This prefactor leads to a modified Bekenstein-Hawking entropy. The semiclassical Bekenstein-Hawking area law connecting the entropy to the horizon area is altered. As obtained in other approaches [9, 10, 12], the leading correction is found to be logarithmic while the nonleading one is a series in inverse powers E-mail: [email protected] E-mail: [email protected]